Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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32 views

Constructing vector topologies (TVS's)

Consider the following theorem extracted from "An Introduction to Functional Analysis" by Charles Swartz (1992): Theorem 1: Let $X$ be a vector space. Let $\mathcal{U}$ be a family of subsets of $...
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2answers
30 views

Eigenvalues of a Matrix over the Complex Numbers

I am looking at this question at the moment: If M is a N×N matrix over ℂ, with $M_{nm}$:=$\frac{N+1}{2}$ if n=m, and $M_{nm}:=\frac{1}{e^{\frac{2\pi(m-n)}{N}}-1}$ if n≠m, then show that the ...
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3answers
31 views

How do I find the normal vector at point $p$ on a cylinder $x^2+y^2=1$?

How do I find the normal vector at point $p$ on a cylinder $x^2+y^2=1$? I would find this normal vector on point $p$ with any graphic of a function like $(-z_x,-z_y,1)$, but in this case I have no $z$ ...
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31 views

Stokes theorem on oriented curve of form $(x+y+z)dx + x^2dy+xyzdz$

This is the formulation, which is not clear to me entirely. Let $S$ be the upper unit half-sphere (this probably means the set $\{(x,y,z)|x^2+y^2+z^2=1, z\geq 0\}$) in the right half-plane plane(I ...
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1answer
61 views

Derivative of matrix exponential at $0$

I have to show that the derivative of 'the matrix exponential' $exp: \mathbb{C}^{n\times n}\mapsto\mathbb{C}^{n\times n}$ at the zero matrix $0$ is $id_{C^{n\times n}}$, i.e. $exp(0)=id$. The above ...
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1answer
102 views

Showing that the omega and the alpha limits are disjoint or have just one common point

Let $f:\mathbb R^2\rightarrow \mathbb R^2$ be a $C^1$ function and $x'=f(x)$. Suppose that there are finites points $x_i\in \mathbb R^2$, such that $f(x_i)=0$. Given $y$, such that $f(y)\neq 0$, and ...
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2answers
71 views

Proofing that the exponential function is continuous in every $x_{0}$

Given: $$\exp: \mathbb{R} \ni x \mapsto \sum_{k=0}^{\infty } \frac{1}{k!} x^{k} \in \mathbb{R}$$ also $e = \exp(1)$. For all $x \in \mathbb{R}$ with $\left | x \right | \leq 1$: $$\left | \exp(x) -...
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1answer
37 views

Find an example for a bounded function $f$ where the graph of $f$ marked by ${E=(x,f(x))}$ is not of volume 0.

Find an example for a bounded function $f$ where the graph of $f$ marked by ${E=(x,f(x))}$ is not of volume 0. I tried to think about some variation of dirichlet function but the I saw that you could ...
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1answer
33 views

Let $\lVert\cdot\rVert_1,\lVert\cdot\rVert_2$ be norms on vector space $X$. Prove that they generate the same topology iff they are equivalent. [duplicate]

Note that by "generate the same topology" we mean that any set that is open with respect to $\lVert\cdot\rVert_1$ is also open with respect to $\lVert\cdot\rVert_2$ and vice versa. By "equivalent" we ...
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1answer
60 views

Proving that the exponential function is continuous

We aren't allowed to use many tricks such as difference quotient / integral calculus... Prove that $\exp$ is continuous at $x_{0}=0$ .....................................................................
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1answer
44 views

Where $\{q_n\}=\mathbb Q$ and $f_n:[q_n-2^{-n-1},q_n+2^{-n-1}]\to[0,\infty)$ with $\int f_n\,d\lambda=1$, show $\sum_{n=1}^\infty f_n<\infty$ a.e.

That is: Let $\mathbb Q=\{q_n\}_{n\in\mathbb N}$ be an enumeration of the rationals. Let $f_n$ be a nonnegative Borel measurable function supported on $q_n\pm 2^{-n-1}$ with $\int f_n\,d\lambda =1$, ...
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1answer
29 views

How can I prove this inequality involving the exponential function?

Given $$\exp: \mathbb{R} \ni x \mapsto \sum_{k=0}^{\infty } \frac{1}{k!} x^{k} \in \mathbb{R}$$ also $e = \exp(1)$. For all $x \in \mathbb{R}$ with $\left | x \right | \leq 1$: $$\left | \exp(x) - ...
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0answers
20 views

Newton's method for nth roots of complex numbers

Is it possible to use Newton's method to compute roots of complex numbers, say $\sqrt[n]{a+ib}$ to any desired accuracy? If yes,for what initial values will converge?
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2answers
28 views

Find stationary points of the function $f(x,y) = (y^2-x^4)(x^2+y^2-20)$

I have problem in finding some of the stationary points of the function above. I proceeded in this way: the gradient of the function is: $$ \nabla f = \left( xy^2-3x^5-2x^3y^2+40x^3 ; x^2y+2y^3-x^4y-...
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1answer
29 views

Question on Inequality from Bartle's Elements of Integration: Riesz Fischer Theorem

I am puzzled how did Bartle get $$|g_k|\leq\sum_{j=k}^\infty |g_{j+1}-g_j|$$ (second last line)? I tried using Triangle Inequality and ended up with one extra term: $$\begin{align*} |g_k|&=|g_k-...
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1answer
34 views

extension of Cauchy–Schwarz inequality

I want to show $\sum_{i,j}a_{i,j}b_{i,j}$<=$\sum_{i,j}a_{i,j}\sum_{i,j}b_{i,j}$, can I use Cauchy–Schwarz inequality to do that? Or is there any conditions needed to show the inequality?
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2answers
69 views

Construct a harmonic function that appears to be discontinuous on the unit circle.

Construct a harmonic function $u$ in $D(0,1)$ that satisfies $$ lim_{r \to 1^-}u(re^{i\theta}) = \begin{cases} 1 & 0 < \theta < \pi \\ 0 & \pi < \theta < 2\pi \...
2
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0answers
56 views

Are there any “default” properties which hold for almost all topological spaces in analysis?

What is the simplest/most general commonly used (type of) topological space in analysis? For instance, every example I can think of in analysis is first-countable. I don't think it can be metric ...
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0answers
12 views

Let $Q=[0,1]\times[0,2]$. Find the rotation matrix $(A)$for the angle $\frac{\pi}{4}$ upon this set. Is $A(Q)$ measurable and a elementary set?

Let $Q=[0,1]\times[0,2]$. Find the rotation matrix $(A)$for the angle $\frac{\pi}{4}$ upon this set. Is $A(Q)$ measurable and a elementary set? First off, I know that $A$ is a linear map, and a ...
4
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1answer
30 views

$\mathcal{l}^1$ is not complete for the norm $\|\cdot\|_\infty$

Let $\mathcal{l}^\infty = \{ (u_n) | u_n \in \mathbb{R}$ and $sup_{n \in \mathbb{N}}|u_n| < \infty \}$ and $\mathcal{l}^1 = \{ (u_n) | u_n \in \mathbb{R}$ and $\sum_{n=1}^{\infty} |u_n| < \infty ...
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1answer
12 views

Functions composition commutativity

I have to prove that $\circ$ is not, in general, a commutative operation of Funct(X,X). My approach: Let X be a set, $a,b\in X$, $a\neq b$ constants. Let $i,j \in Funct(X,X)$ with $i:X \to X,\text{ } ...
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1answer
25 views

Prove or disprove: $\lVert T\rVert=\sup_{\lVert x\rVert=1}|\langle Tx,x\rangle|$, where $H$ is a Hilbert space and $T$ is bdd linear operator.

Edit: To clarify, note that $T:H\to H$. This is a problem on an old preliminary exam in Analysis that I'm working through to prep for my own prelim. My initial thought was to disprove it, but I can'...
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1answer
31 views

Good, simple reference for Riesz-Fischer Theorem.

I am looking for a good, simple reference for the proof of Riesz-Fischer Theorem ($L^p$ spaces are complete). An example of a not so good reference in my opinion is Royden, where he uses "rapidly ...
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2answers
52 views

Preimage of a function

I'm having difficulties with the notion of preimage, specifically with this example: Let $A$ be a subset of $[0, 1]$. We define $$f(x) = \begin{cases} x, & x \in A; \\ -x, & x \in [0, 1] \...
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3answers
47 views

Finding limit of this function

Given function: f(x) = $\sqrt{x^2+x}-x$ I used 3th binomial formula and brought it to this form: $\frac{x}{\sqrt{x^2+x}+x}$ But now no idea how to get limit of this (goes to ∞). By testing I know ...
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3answers
182 views

Evaluate the integral $ \int_0^{+\infty} \frac{\sin(x^2)}{x^4+1} dx $ using the residue method

I have a problem in evaluating the integral above. So far I've proceeded in this way. We have an even function, so: $$ \int_0^{+\infty} \frac{\sin(x^2)}{x^4+1} dx = \frac{1}{2} \int_{-\infty}^{+\...
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1answer
30 views

Infinite union of countable sets proof.

I understand how to prove that the union of 2 countable sets is countable. I then began to think we can use induction to say that the countable union of countable sets are also countable. However my ...
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1answer
61 views

Why is $E$ measurable

I have some queries about this proof (towards the last part of this text). 1) Firstly, why is $E=\{x\in X:g(x)<\infty\}$ measurable? 2) Secondly, why is $\mu(X\setminus E)=0$? Is it because $\|g\|...
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0answers
29 views

Square root of several variables analytic function

The question is as follow: Let $H\subset \mathbb{C}^n$ be an simply-connected region. If $f$ is a nowhere vanishing analytic function on $H$, with $f(z)>0$ for all $z\in H\cap\mathbb{R}^n$, ...
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0answers
22 views

Show that change of variable theorem holds for $\cup A_i$

Problem assumes the following theorem $\mathtt{Theorem}$ Let A be an open subset of $\mathbb R^n$ and $\phi:A \to \mathbb R^n$ a one-to-one continuously differentiable map whose Jacobian $J\phi$ ...
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3answers
55 views

Limit of type 0/0 without L'Hopital

I am trying to figure out the limit as $h \rightarrow 0$ of $$\frac{x\sqrt{x+h+1}-x\sqrt{x+1}}{h(x+h)}$$ without using l'Hopital's rule. I have tried to extend the fraction by the conjugate of ...
0
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1answer
26 views

$f'_n(x)$ is bounded and $f_n(x) \to 0$ for each x, then $f_n(x) \to 0$ uniformly

I want to show the question If $f_n(x)$ is differentiable on [a.b] with $|f'_n(x)|<10$ for all n and if $f_n(x) \to 0$ at each x, then $f_n(x) \to 0$ uniformly. I think I should use triangle ...
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4answers
52 views

$\lim\limits_{n\to \infty}\sum_{k=1}^{\infty}2^{-k}\sin(k/n)=0$

I want to show that $$\lim\limits_{n\to \infty}\sum_{k=1}^{\infty}2^{-k}\sin(k/n)=0$$ I first thought if I can change the order of limit, it can be easy to show that. But I found that there ...
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0answers
16 views

Finding a branch of the complex logarithmic function $\log(1-z).$

I have a question that asks me to find the holomorphic branch $L(1 − z)$ of $\log(1 − z)$ valid in the cut-plane $z \in \mathbb{C}\setminus [1, ∞)$ and such that $L(1) = 0.$ We have defined the ...
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0answers
19 views

Convergance of the average of a convergant complex sequence

So this is in exercise 3.14 (Neat!) in Baby Rudin, which I have found quite a simple and obvious proof for, however when I checked the answers the proofs I found were quite complicated so now I am a ...
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0answers
35 views

Almost found the limit points of this set

I want to find all limit points of the given set, and I think I almost got it. $M=\left\{\frac{x}{2^y} \mid x, y\in ℕ , x \leq y\right\}$ also $M \subset ℝ$. We say that: $\forall \delta > 0,\...
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2answers
43 views

Partial fraction decomposition of $\pi\cdot \tan(\pi z)$

Evaluate the partial fraction decomposition of $\pi \tan(\pi z)$ $$2\pi \tan(\pi z)=\cot\left(\frac{\pi}{2}-\pi z\right)-\cot\left(\frac{\pi}{2}+\pi z\right)$$ $$=\frac{2}{1-2z}+\sum_{k=1}^\infty \...
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0answers
24 views

How do I describe the analytic completion to the algebraic closure of $\mathbb{F}_2$?

Is it possible? I'm using the algebra generated by the set $\{0, 1^r\}$ for all fractions $r$ as my representation of the algebraic closure to $\mathbb{F}_2$. I can't seem to find a metric on it. ...
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0answers
33 views

Does $\min\{x_1x_3+x_2^2 | x_1^2+x_2^2+x_3^4 = 4\}$ has an optimal solution?

Does $\min\{x_1x_3+x_2^2 | x_1^2+x_2^2+x_3^4 = 4\}$ has an optimal solution? I think continuous function over closed and bounded domain has an optimal solution but I am not sure. Can anyone give me ...
0
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2answers
43 views

Finding all limit points of a set

How can I find all the limit points of this set? $S=\left\{\frac{x}{2^y} \mid x, y\in ℕ , x \leq y\right\}$ with $S \subset ℝ$. Could this be proved if I showed that $∀δ > 0, \exists z ∈ S\text{...
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1answer
42 views

Urysohn's extension theorem

Currently I am working my way through Ernest Michael's first article on continuous selections. Here, Urysohn's extension theorem is stated as follows: For a $T_1$-space, the following properties ...
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1answer
30 views

How to find all limit points of this set?

How to find all limit points of this set? $S=\left\{\frac{x}{y} \mid x, y\in ℕ , x \leq y\right\}$ also $S \subset ℝ$. Is the way to proof this done same way as it is for sequences? I have never ...
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0answers
154 views

Show that map is norm preserving and determine Eigenvalues [duplicate]

Can someone of you give me a solution for this? Let $N\in \mathbb N$. a) We define the map $\mathfrak F:(\mathbb C^N, ||\cdot||_2)\to(\mathbb C^N, ||\cdot||_2)$ by $$(\mathfrak F(x))_k := \frac{...
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2answers
36 views

Is $f_n=x^n$ weakly convergent in $(\mathscr C[0,1],\lVert\cdot\rVert_\infty)$?

This is part of an old preliminary exam in Analysis I am working through. For earlier parts of the problem I have already shown that $f_n$ does not converge in $(\mathscr C[0,1],\lVert\cdot\rVert_\...
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0answers
15 views

Alexandrov Maximum Principle and $W^{2}_p$ estimates

I'm reading an article of N. V. Krylov: About an example of N. N. Ural'tseva and weak uniqueness for elliptic operators, Nonlinear partial differential equations and related topics, 131–144. This ...
1
vote
1answer
28 views

Prove that exists a linear continuous functional satisfying…

Let $E$ be a normed space over the field of real numbers. I have to prove that given two convex sets $A$, $B$ in $E$, with positive distance between then, there exists a linear continuous functional ...
2
votes
1answer
43 views

Convergent + divergent $\to$ divergent

Given sequences $(x_n)$, convergent, but $(y_n)$ is divergent, then $(x_n + y_n)$ is divergent. I am confident that it is true, but having trouble getting the formalities correct. I have tried proof ...
2
votes
1answer
53 views

Is it possible to construct such a function in analytical form?

Suppose $f\left(f\left(x\right)\right)=\sin(x)$ Is it possible to find $f$ in closed form, or any other forms so as to visualize $f(x)$ on $x\in[-\pi,\pi]$? Is it possible to prove the existence and ...
0
votes
1answer
31 views

Does an unbounded gradient implies a non-vanishing hessian determinant

Let $\Phi:\mathbb{R^{n-1}\to R}$ be a vector function. Suppose that exists a set $S_1\subset \mathbb R^{n-1}$ on which $G=\nabla\Phi$, the gradient of the function is unbounded. Does that imply ...
1
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4answers
57 views

The continuity of function's restrictions implies the continuity of function.

Let be $X \subset F_1 \cup F_2$, where $F_1$ and $F_2$ are closed. If the function $f\colon X \longrightarrow \mathbb{R}$ is such that $f|_{X \cap F_1}$ and $f|_{X \cap F_2}$ are continuous, so prove ...